Photometric Monitoring of the First Eclipsing Binary Be Star: V658 Car

Abstract

V658 Car is the first known eclipsing binary system involving a classical Be star and an sdOB companion, offering a unique opportunity to study disk physics and binary interactions in unprecedented detail. From TESS data and multi-color observations from the comissão para a colaboração entre profissionais e amadores collaboration, we analyze the system’s color–magnitude diagram and compare it with radiative transfer models that include the Be star, its circumstellar disk, and the sdOB companion. While the stellar eclipses are well reproduced, two features observed in the multi-color photometry challenge the current modeling paradigm: the discrepancy between the observed reddening and the modeled blueing during the first attenuation phase and the complete lack of modeled attenuation around the second stellar eclipse. These issues highlight the need for more sophisticated modeling approaches to capture the complex interplay between disk opacity and binary dynamics.

1. Introduction

Classical Be stars are rapidly rotating main-sequence stars that, for reasons not yet fully understood, undergo episodic mass loss that leads to the formation of a circumstellar disk [1]. While the physical mechanism behind these outbursts remains uncertain, non-radial pulsations are often considered the most promising explanation [2,3]. Regardless of the exact trigger, fast rotation is clearly a critical factor, either by initiating the mechanism directly or by facilitating it by lowering the energy barrier required to expel material. Consequently, understanding the origin of Be stars is intrinsically tied to understanding the origin of their spin-up.

Among the various formation scenarios proposed, binary evolution is of particular relevance to this study. In this scenario, the initially more massive star evolves more quickly and transfers its outer layers to its lower-mass companion via Roche lobe overflow. The result is the stripped core of the donor and a rejuvenated, rapidly rotating Be star that has gained both mass and angular momentum [4]. After mass transfer ceases, the donor star is initially observed as a bloated star [5] before it eventually begins to shrink and evolve into a subdwarf (sdOB), which is smaller than the Sun. Over the past decade, approximately 20 such systems [6], known as Be + sdOB binaries, have been identified. However, their detection is challenging: although the hot subdwarf companion is typically hotter than the Be star, it is generally much fainter and therefore difficult to observe directly (e.g., $\kappa \mathrm{Draconis}$ [7]).

In this context, V658 Car stands out as a benchmark example of a Be + sdOB system. Its companion is unusually luminous, comparable to the Be star, and the nearly edge-on orientation of the system makes it the first known classical Be eclipsing binary. This geometry provides a rare opportunity to study disk physics in Be binaries in unprecedented detail. The system was first identified as binary by [8] and as an eclipsing Be Binary by [9]. Recently, [10] classified it as an eclipsing Ae + B binary, with similar stellar radii ($R_{\mathrm{Ae}}=1.64\pm 0.04{\mathrm{R}}_{\odot }$ and $R_{\mathrm{B}}=1.46\pm 0.04{\mathrm{R}}_{\odot }$) and effective temperatures ($T_{\mathrm{Ae}}=9.7\pm 0.5\mathrm{kK}$ and $T_{\mathrm{B}}=12.7\pm 0.7\mathrm{kK}$). This interpretation is now considered unlikely due to the presence of helium absorption lines in the sdOB spectrum [11].

Recently, [11] (hereafter Paper I) re-analyzed V658 Car, simultaneously fitting the polarimetry in the BRVI bands, the TESS light curve, the ${\mathrm{H}}_{\alpha }$ equivalent width, and the spectral energy distribution (SED) from the ultraviolet to the near-infrared. Based on this comprehensive dataset and improved modeling, Paper I demonstrated that V658 Car is consistent with a Be + sdOB configuration, or at least that its companion is currently contracting and will soon enter the subdwarf phase.

Crucially, the nearly edge-on, double-lined (SB2) nature of the system enabled the authors to determine the component masses with high precision. Using a TESS (Transiting Exoplanet Survey Satellite, [12]) light curve, they also constrained the radii of both components to high accuracy and uncovered new details about disk behavior that cannot be addressed in less favorable systems.

Paper I successfully reproduced the observed polarization, the ${\mathrm{H}}_{\alpha }$ emission level, the ultraviolet and optical spectral energy distribution (SED), and both stellar eclipses in the TESS light curve. However, the results are not without shortcomings. The authors encountered difficulties in reproducing the ${\mathrm{H}}_{\alpha }$ equivalent width variability, the infrared portion of the SED, and the observed stellar attenuations (as discussed in Section 4 and Section 5). Although their adopted model explains the overall system properties reasonably well, it does not fully reproduce all observed features.

In this paper, we present a complementary analysis by exploring the positional variation of V658 Car in the B- and V-band color–magnitude diagram (CMD) during the stellar attenuations and eclipses. The CMD was obtained in collaboration with the comissão para a colaboração entre profissionais e amadores (COPA) from the Brazilian Astronomical Society.

2. Observational Data

TESS observed V658 Car over five sequences of approximately 30 days each: sector 10 in 2019, sectors 36 and 37 in 2021, and sectors 63 and 64 in 2023. With a high cadence (2 min) and a signal-to-noise ratio of about 13,000, these observations enabled a detailed analysis of the eclipses. However, they were obtained in a single band in the range 600–$1000\mathrm{nm}$.

In partnership with the COPA collaboration, we obtained multi-color absolute photometry in the Johnson B and V bands. Observations were conducted at eight different sites, with a cadence of one observation per hour during stellar eclipses and one observation per day throughout the remainder of the orbital cycle, each of them consisting of ten images. For each observation, we used the mean flux as the representative flux and the standard deviation as the associated uncertainty. V658 Car was observed 335 times in the B filter and 494 times in the V filter between March and July 2022, achieving a precision of 10 and 12 millimagnitudes, respectively.

We employed standard data reduction techniques, including aperture photometry, bias and dark subtraction, flat-field correction, and the removal of bad frames (e.g., saturated images). We also highlight that one of the eight sites produced measurements with a systematic offset of 50 millimagnitudes; this offset was subtracted from the final dataset. To perform absolute photometric calibration, we used the AstroImageJ V5.0.3.00 software created by Karen Collins, Kevin Eastridge and John Kielkopf (University of Louisville, Louisville, KY, USA) [13], specifically Equations C1 and C2, which describe the weighted average used. We selected four relatively constant stars (as seen by TESS and indicated by the spectral type) as field stars: HD 303181, HD 303097, HD 303094, and TYC 8613-495-1. To produce a cleaner color–magnitude diagram, thereby avoiding crowding while allowing for a one-to-one comparison of the B and V observations, we binned the data into phase intervals of 0.025, 0.009, and 0.003 for the flat portion, stellar attenuation, and stellar eclipses, respectively. Within each bin, we computed the average values weighted by the observational uncertainties. The original, unbinned data are shown in Figure 1, while Figure 2 shows a CMD diagram with the binned data.

3. Modeling Overview

For the sake of completeness, we briefly describe the model developed for this system and reported by Paper I. It consists of three components: the Be star, its circumstellar disk, and the hot subdwarf companion. The disk was assumed to be axisymmetric around the Be star and truncated at the Roche lobe radius. In Figure 1 of Paper I [11], the authors show a schematic view of the system, with the respective definition of the phases (where $\mathrm{phase}=0$ corresponds to the eclipse of the sdOB star). We adopt their best-fit parameters, which is summarized in

Table 1. Model input parameters corresponding to the best-fit solution obtained in Paper I [11]. These parameters were kept fixed during the modeling.

Parameter	Unit	Component
Component		Be star	Companion
Mass	${\mathrm{M}}_{\odot }$	$4.45$	$0.53$
Semi-Major Axis	${\mathrm{R}}_{\odot }$	$7.8$	$64.92$
Polar Radius	${\mathrm{R}}_{\odot }$	2.40	0.97
Equatorial Radius	${\mathrm{R}}_{\odot }$	3.37	0.97
Polar Temperature	kK	18.0	21.0
Rotational Velocity	km/s	452	23
Luminosity	${\mathrm{L}}_{\odot }$	347	163
Component		Be disk
m	–	2.0
$\beta$	–	1.5
Density at the base of the disk	$\mathrm{g}{\mathrm{cm}}^{-3}$	$6.01\times {10}^{-11}$
Disk External Radius	${\mathrm{R}}_{\odot }$	41
Component		System
Ellipticity	–	0
Period	d	32.18534
Ephemeris Time	BJD	2,459,545.368
Inclination	°	88.6
Distance	pc	663
Interstellar Color Excess	mag	$0.12$

.

The disk structure was based on the viscous decretion disk (VDD) formulation originally proposed by [14]. In this scenario, material ejected from the Be star surface is redistributed outward as a result of viscous torques, forming a Keplerian disk. The disk is geometrically thin, with most material concentrated in the equatorial plane, and vertically supported by hydrostatic equilibrium. Its vertical extent, or scale height, is described by a power law as a function of the radius, r,

$$H\left(r\right)=H_0{\left({\displaystyle \frac{r}{R_{\mathrm{eq}}}}\right)}^{\beta },$$

(1)

where $\beta$ is the flaring parameter, $H_0$ is the scale height at the base of the disk, and $R_{\mathrm{eq}}$ is the Be equatorial radius. A commonly used and useful approximation for the radial dependence of the density is (see, e.g., [15])

$$\rho \left(r\right)={\displaystyle \frac{{\rho }_0}{H\left(r\right)}}{\left({\displaystyle \frac{r}{R_{\mathrm{eq}}}}\right)}^{-m},$$

(2)

where m is the disk density slope and ${\rho }_0$ is the density at the base of the disk. Under the isothermal approximation [16] and assuming that the disk was fed at a constant rate for a long time [17], typical parameters are $m=2$, $\beta =1.5$, and $H_0=c_{\mathrm{s}}{R}_{\mathrm{eq}}/V_{\mathrm{orb}}$, where $c_{\mathrm{s}}$ is the sound speed (computed using 72% of the star’s polar temperature; [18]) and $V_{\mathrm{orb}}$ is the local orbital velocity.

The presence of a binary companion introduces modifications to this idealized structure. As explained in [19], the additional gravitational potential from the sdOB companion reduces the disk’s vertical extent in the vicinity of the companion. The density profile should also become flatter (lower m) due to the mass accumulation effect, where the companion steals angular momentum from the disk, requiring additional torque for it to grow and thus more mass on the outskirts. To loosely simulate these effects, Paper I constructed an alternative model with $m=1.5$, reflecting a denser outer disk. However, both density prescriptions produce effectively identical results. For clarity and simplicity, we have chosen to present only the $m=2$ solution in the photometric analysis.

The modeling is executed in two stages. In the first, the 3D Monte Carlo radiative transfer code HDUST [18,20] is used to simulate the thermal structure of the disk, by computing electron temperatures and hydrogen-level populations, and the subsequent continuum emission. In its current version, HDUST can self-consistently solve the radiative equilibrium equation in the disk by considering a single stellar source. The companion star is thus excluded from this stage due to current code limitations, which prevent the inclusion of multiple radiation sources. This is an ongoing effort that will be reported in future publications.

In the second stage, Paper I employs a ray-tracing code specifically designed for this project to compute the observed flux from each stellar component, incorporating disk absorption and occultation effects as a function of the orbital phase. For the stellar photospheric spectra, the authors adopt models from Coelho [21] for the Be star and Lanz & Hubeny [22] for the hot subdwarf companion. The final synthetic spectrum includes the continuum contributions from the Be star, its disk, and the companion, properly accounting for any phase-dependent eclipses or attenuations, which are corrected for interstellar reddening using the color excess.

4. Results

Data from TESS sectors 36 and 37 1 are plotted in Figure 2. The overall shape of the TESS light curve for V658 Car (bottom panels) reveals two short, deep, and sharp stellar eclipses (hereafter, the first and second stellar eclipses, with orbital phases 0 and 0.5, respectively) in the center of two broad and shallower dimmings that do not resemble normal eclipses (hereafter, the first and second attenuations, following the above phase convention). The first stellar eclipse is about 50% deeper than the second (relative to the adjacent brightness level), and they do not present flat-bottom features. For reference, the model predicts a relative contribution in the V band of 63.5%, 30%, and 6.5% for the Be star, the companion, and the disk emission, respectively. The overall behavior in the B and V light curves in Figure 1 agrees qualitatively well with the TESS data.

The broad attenuations show notable differences from one another. The second one is highly symmetric around phase 0.5, exhibits a slow rate of brightness variation with a smooth transition from the baseline, and spans a wide range in terms of the phase (approximately from 0.3 to 0.7). In contrast, the first attenuation has a much larger rate of variation and features sharp edges. More importantly, the first attenuation is highly asymmetric, with the ingress 2 being narrower and slightly shallower than the egress. Similar to observations in other eclipsing binaries, such as Algol-like stars [23], ref. [10] linked the broad attenuations to absorption by circumstellar material.

In the color–magnitude diagram (CMD, Figure 2), the flat portion of the light curve (shown in cyan), where there is no eclipse or attenuation, is characterized by $B\approx V\approx 9.10\mathrm{mag}$, consistent with the values measured by Hauck [10], where $B=9.08$ mag and $V=9.11$ mag. The first stellar eclipse (in red) appears as a fading and reddening from $V=9.2$ and $B-V=0.02$ mag to $V=9.4$ and $B-V=0.06$ mag as the companion (hotter) star is being eclipsed. The first attenuation is more intricate; during this part (in green), the system first becomes redder and dimmer, reaching $V=9.18$ and $B-V=0.04$, and then it just becomes bluer, reaching $B-V=0.02$. The models predict a different scenario, although a similar attenuation is noticed, and the system becomes bluer–opposite to what is observed.

During the second attenuation (in green), the system becomes bluer and dimmer ($V\approx 9.16\mathrm{mag}$ and $B-V\approx -0.01\mathrm{mag}$) when the companion is in front of the Be star. This would be consistent with a dimming of the Be star, as it represents the redder component. We do not have enough coverage of the second stellar eclipse to draw precise conclusions. However, as the Be star would be further dimmed, we would expect the system to become even bluer.

5. Discussion

Similarly to Paper I, where the model successfully reproduced some observational features but failed to match others, the same pattern is evident in Figure 2.

The first notable point is that the color of our best-fit solution during the flat portion of the light curve is reasonably close to the observed values, within the standard deviation from a perfect match.

The primary issue with the models arises during the first attenuation phase. Observations show that the system becomes redder during this period, which is broadly expected for two reasons: (1) the contribution from the bluer component decreases, and (2) a circumstellar disk, being much cooler than the stars, should generally redden the system. However, our models predict the opposite trend, with the system becoming bluer. This is likely due to the wavelength-dependent nature of bound-free opacity (see Figure 1 of [24]) in the disk: in the Balmer continuum, the disk becomes more optically thick at longer wavelengths, which dominates the color change and leads to a net reddening effect. A similar phenomenon was reported in [17]. The authors examine the CMD during the build-up phase, during which the star actively loses mass and the disk grows. This scenario is comparable, as disk attenuation resembles the build-up process. Exploring CMDs with alternative color indices, such as U–V, may help clarify the true dynamics of disk absorption.

The first stellar eclipse is well reproduced by our models, provided that the first attenuation discrepancy is ignored. Similar to observations, the model gets dimmer by approximately 0.2 mag in the V band and redder by about 0.03 mag, indicating that the stellar parameters are likely accurate and that the discrepancies arise mainly from the disk solution.

As mentioned in Section 3, our two proposed solutions, $m=2$ and $m=1.5$, yield identical results. This suggests that the current disk model implementation may not be sufficient to resolve the remaining inconsistencies. This conclusion is consistent with Paper I, where the authors also identified a few inconsistencies. The main difference, however, is that most of the issues discussed in Paper I are expected to be resolved with the development of more refined models that account for the complex interplay between disk opacity and binary dynamics. In contrast, this paper presents two features in the multi-color photometry that such refinements are unlikely to resolve, which therefore pose key challenges for any future modeling efforts: the discrepancy between the observed and modeled behavior during the first attenuation and the absence of any dimming in the models during the second attenuation.